To determine the average rate at which the object falls during the first 3 seconds, we can use the concept of average velocity, which is the change in height divided by the change in time.
### Step-by-Step Solution
1. Understand the height function:
The height function [tex]\( h(t) \)[/tex] describes the height of the object at any time [tex]\( t \)[/tex]. We are given:
[tex]\[
h(t) = 300 - 18t^2
\][/tex]
2. Calculate the height at [tex]\( t = 0 \)[/tex]:
When [tex]\( t = 0 \)[/tex] (the moment the object is dropped), the height is:
[tex]\[
h(0) = 300 - 18 \cdot 0^2 = 300
\][/tex]
3. Calculate the height at [tex]\( t = 3 \)[/tex]:
When [tex]\( t = 3 \)[/tex] seconds, the height is:
[tex]\[
h(3) = 300 - 18 \cdot 3^2 = 300 - 18 \cdot 9 = 300 - 162 = 138
\][/tex]
4. Determine the change in height:
The change in height over the 3 seconds is the difference between the height at [tex]\( t = 0 \)[/tex] and [tex]\( t = 3 \)[/tex]:
[tex]\[
\Delta h = h(3) - h(0) = 138 - 300 = -162
\][/tex]
5. Calculate the average rate of fall:
The average rate of fall (average velocity) over the 3 seconds is the change in height divided by the time interval, [tex]\( \Delta t = 3 \)[/tex] seconds:
[tex]\[
\text{Average rate of fall} = \frac{\Delta h}{\Delta t} = \frac{h(3) - h(0)}{3} = \frac{-162}{3} = -54 \, \text{feet per second}
\][/tex]
Thus, the correct expression to determine the average rate at which the object falls during the first 3 seconds is:
[tex]\[
\boxed{\frac{h(3) - h(0)}{3}}
\][/tex]
